Vibratory flowmeter and method for meter verification

ABSTRACT

A vibratory flowmeter ( 5 ) for meter verification is provided, including meter electronics ( 20 ) configured to vibrate the flowmeter assembly ( 10 ) in a primary vibration mode using the first and second drivers ( 180 L,  180 R), determine first and second primary mode currents ( 230 ) of the first and second drivers ( 180 L,  180 R) for the primary vibration mode and determining first and second primary mode response voltages ( 231 ) generated by the first and second pickoff sensors ( 170 L,  170 R) for the primary vibration mode, generate a meter stiffness value ( 216 ) using the first and second primary mode currents ( 230 ) and the first and second primary mode response voltages ( 231 ), and verify proper operation of the vibratory flowmeter ( 5 ) using the meter stiffness value ( 216 ).

TECHNICAL FIELD

The present invention relates to a vibratory flowmeter and method, andmore particularly, to a vibratory flowmeter and method for meterverification.

BACKGROUND OF THE INVENTION

Vibrating conduit sensors, such as Coriolis mass flowmeters andvibrating densitometers, typically operate by detecting motion of avibrating conduit that contains a flowing material. Propertiesassociated with the material in the conduit, such as mass flow, densityand the like, can be determined by processing the measurement signalsreceived from the motion transducers associated with the conduit. Thevibration modes of the vibrating material-filled system generally areaffected by the combined mass, stiffness and damping characteristics ofthe conduit and the material contained therein.

A typical dual-driver, or multiple input, multiple output (MIMO)Coriolis mass flowmeter includes one or more conduits, or flow tubes,that are connected inline in a pipeline or other transport system andconvey material, e.g., fluids, slurries, emulsions, and the like, in thesystem. Each conduit may be viewed as having a set of natural vibrationmodes, including for example, simple bending, torsional, radial, andcoupled modes. In a typical dual-driver Coriolis mass flow measurementapplication, a conduit is excited in one or more vibration modes as amaterial flows through the conduit, and motion of the conduit ismeasured at points spaced along the conduit. Excitation is typicallyprovided by two actuators, e.g., electromechanical devices, such asvoice coil-type drivers, that perturb the conduit in a periodic fashion.Mass flow rate may be determined by measuring time delay or phasedifferences between motions at the transducer locations. Two suchtransducers (or pickoff sensors) are typically employed in order tomeasure a vibrational response of the flow conduit or conduits, and aretypically located at positions upstream and downstream of the actuator.The two pickoff sensors are connected to electronic instrumentation. Theinstrumentation receives signals from the two pickoff sensors andprocesses the signals in order to derive a mass flow rate measurement ora density measurement, among other things.

It is a problem that the one or more conduits may change with time,wherein an initial factory calibration may change over time as theconduits are corroded, eroded, or otherwise changed. As a consequence,the conduit stiffness may change from an initial representativestiffness value (or original measured stiffness value) over the life ofthe vibratory flowmeter.

Mass flow rate ({dot over (m)}) may be generated according to theequation:{dot over (m)}=FCF*[Δt−Δt _(o)]  (1)

The Flow Calibration Factor (FCF) is required to determine a mass flowrate measurement ({dot over (m)}) or a density measurement (ρ) of afluid. The (FCF) term comprises a Flow Calibration Factor and typicallycomprises a geometric constant (G), Young's Modulus (E), and a moment ofinertia (I), wherein:FCF=G*E*I  (2)The geometric constant (G) for the vibratory flowmeter is fixed and doesnot change. The Young's Modulus constant (E) likewise does not change.By contrast, the moment of inertia (I) may change. One way to track thechanges in moment of inertia and FCF of a vibratory flowmeter is bymonitoring the stiffness and residual flexibility of the flowmeterconduits. There are increasing demands for ever better ways to trackchanges in the FCF, which affect the fundamental performance of avibratory flowmeter.

What is needed is a technique to track the FCF in a dual-driverflowmeter to verify the performance of the flowmeter with improvedprecision.

SUMMARY OF THE INVENTION

A vibratory flowmeter for stiffness verification is provided accordingto an embodiment of the Application. The vibratory flowmeter for meterstiffness verification includes a flowmeter assembly including one ormore flowtubes and first and second pickoff sensors; first and seconddrivers configured to vibrate the one or more flowtubes; and meterelectronics coupled to the first and second pickoff sensors and coupledto the first and second drivers, with the meter electronics beingconfigured to vibrate the flowmeter assembly in a primary vibration modeusing the first and second drivers, determine first and second primarymode currents of the first and second drivers for the primary vibrationmode and determining first and second primary mode response voltagesgenerated by the first and second pickoff sensors for the primaryvibration mode, generate a meter stiffness value using the first andsecond primary mode currents and the first and second primary moderesponse voltages, and verify proper operation of the vibratoryflowmeter using the meter stiffness value.

A method for meter verification method for a vibratory flowmeter isprovided according to an embodiment of the Application. The methodincludes vibrating a flowmeter assembly of the vibratory flowmeter in aprimary vibration mode using a first driver and at least a seconddriver; determining first and second primary mode currents of the firstand second drivers for the primary vibration mode and determining firstand second primary mode response voltages of first and second pickoffsensors for the primary vibration mode; generating a meter stiffnessvalue using the first and second primary mode currents and the first andsecond primary mode response voltages; and verifying proper operation ofthe vibratory flowmeter using the meter stiffness value.

Aspects

Preferably, the first and second primary mode currents comprisecommanded current levels.

Preferably, the first and second primary mode currents comprise measuredcurrent levels.

Preferably, the second driver is uncorrelated with the first driver.

Preferably, the meter electronics is further configured to compare themeter stiffness value to a predetermined stiffness range, generate averification indication for the vibratory flowmeter if the meterstiffness value falls within the predetermined stiffness range, andgenerate a verification failure indication for the vibratory flowmeterif the meter stiffness value does not fall within the predeterminedstiffness range.

Preferably, the meter electronics is further configured to vibrate theflowmeter assembly in a secondary vibration mode using the first andsecond drivers, determine first and second secondary mode currents ofthe first and second drivers for the secondary vibration mode anddetermine first and second secondary mode response voltages of the firstand second pickoff sensors for the secondary vibration mode, andgenerate the meter stiffness value using one or both of the first andsecond primary mode currents and the first and second primary moderesponse voltages or the first and second secondary mode currents andthe first and second secondary mode response voltages.

Preferably, the meter electronics is further configured to generate ameter residual flexibility value using the first and second primary modecurrents and the first and second primary mode response voltages.

Preferably, the meter electronics is further configured to generate ameter residual flexibility value using the first and second primary modecurrents and the first and second primary mode response voltages,compare the meter residual flexibility value to a predetermined residualflexibility range, and generate a verification indication for thevibratory flowmeter if the meter residual flexibility value falls withinthe predetermined residual flexibility range, and generate averification failure indication for the vibratory flowmeter if the meterresidual flexibility value does not fall within the predeterminedresidual flexibility range.

Preferably, the meter electronics is further configured to vibrate theflowmeter assembly in a secondary vibration mode using the first andsecond drivers, determine first and second secondary mode currents ofthe first and second drivers for the secondary vibration mode anddetermining first and second secondary mode response voltages of thefirst and second pickoff sensors for the secondary vibration mode, andgenerate a meter residual flexibility value using one or both of thefirst and second primary mode currents and the first and second primarymode response voltages or the first and second secondary mode currentsand the first and second secondary mode response voltages.

Preferably, the first driver current and the second driver currentcomprise commanded current levels.

Preferably, the first driver current and the second driver currentcomprise measured current levels.

Preferably, the first response voltage and the second response voltagecomprise substantially maximum response voltages quantified by the firstand second pickoff sensors.

Preferably, the second driver is uncorrelated with the first driver.

Preferably, verifying proper operation of the vibratory flowmetercomprises comparing the meter stiffness value to a predeterminedstiffness range, generating a verification indication for the vibratoryflowmeter if the meter stiffness value falls within the predeterminedstiffness range, and generating a verification failure indication forthe vibratory flowmeter if the meter stiffness value does not fallwithin the predetermined stiffness range.

Preferably, further comprising vibrating the flowmeter assembly in asecondary vibration mode using the first driver and at least the seconddriver, determining first and second secondary mode currents of thefirst and second drivers for the secondary vibration mode anddetermining first and second secondary mode response voltages of firstand second pickoff sensors for the secondary vibration mode, andgenerating the meter stiffness value using one or both of the first andsecond primary mode currents and the first and second primary moderesponse voltages or the first and second secondary mode currents andthe first and second secondary mode response voltages.

Preferably, further comprising generating a meter residual flexibilityvalue using the first and second primary mode currents and the first andsecond primary mode response voltages.

Preferably, further comprising generating a meter residual flexibilityvalue using the first and second primary mode currents and the first andsecond primary mode response voltages, comparing the meter residualflexibility value to a predetermined residual flexibility range,generating a verification indication for the vibratory flowmeter if themeter residual flexibility value falls within the predetermined residualflexibility range, and generating a verification failure indication forthe vibratory flowmeter if the meter residual flexibility value does notfall within the predetermined residual flexibility range.

Preferably, further comprising vibrating the flowmeter assembly in asecondary vibration mode using the first driver and at least the seconddriver, determining first and second secondary mode currents of thefirst and second drivers for the secondary vibration mode anddetermining first and second secondary mode response voltages of firstand second pickoff sensors for the secondary vibration mode, andgenerating a meter residual flexibility value using one or both of thefirst and second primary mode currents and the first and second primarymode response voltages or the first and second secondary mode currentsand the first and second secondary mode response voltages.

BRIEF DESCRIPTION OF THE DRAWINGS

The same reference number represents the same element on all drawings.The drawings are not necessarily to scale.

FIG. 1 shows a vibratory flowmeter for meter verification according toan embodiment of the invention.

FIG. 2 shows meter electronics for meter verification of the vibratoryflowmeter according to an embodiment of the invention.

FIG. 3 is a graph of frequency response showing the effect of residualflexibility.

FIG. 4 represents a vibratory flowmeter having curved flowtubes whereinthe two parallel curved flowtubes are vibrated in a bending mode.

FIG. 5 represents the vibratory flowmeter wherein the two parallelcurved flowtubes are vibrated in a twist (or Coriolis) mode.

FIG. 6 is a flowchart of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention.

FIG. 7 is a flowchart of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention.

FIG. 8 is a flowchart of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1-8 and the following description depict specific examples toteach those skilled in the art how to make and use the best mode of theinvention. For the purpose of teaching inventive principles, someconventional aspects have been simplified or omitted. Those skilled inthe art will appreciate variations from these examples that fall withinthe scope of the invention. Those skilled in the art will appreciatethat the features described below can be combined in various ways toform multiple variations of the invention. As a result, the invention isnot limited to the specific examples described below, but only by theclaims and their equivalents.

FIG. 1 shows a vibratory flowmeter 5 for meter verification according toan embodiment of the invention. The flowmeter 5 comprises a flowmeterassembly 10 and meter electronics 20 coupled to the flowmeter assembly10. The flowmeter assembly 10 responds to mass flow rate and density ofa process material. The meter electronics 20 is connected to theflowmeter assembly 10 via the leads 100 to provide density, mass flowrate, and temperature information over a communication link 26, as wellas other information. A Coriolis flowmeter structure is describedalthough it is apparent to those skilled in the art that the presentinvention could also be operated as a vibrating tube densitometer.

The flowmeter assembly 10 includes manifolds 150 and 150′, flanges 103and 103′ having flange necks 110 and 110′, parallel flowtubes 130 and130′, first and second drivers 180L and 180R, and first and secondpickoff sensors 170L and 170R. The first and second drivers 180L and180R are spaced apart on the one or more flowtubes 130 and 130′. Inaddition, in some embodiments the flowmeter assembly 10 may include atemperature sensor 190. The flowtubes 130 and 130′ have two essentiallystraight inlet legs 131 and 131′ and outlet legs 134 and 134′ whichconverge towards each other at the flowtube mounting blocks 120 and120′. The flowtubes 130 and 130′ bend at two symmetrical locations alongtheir length and are essentially parallel throughout their length. Thebrace bars 140 and 140′ serve to define the axis W and the substantiallyparallel axis W′ about which each flowtube oscillates.

The side legs 131, 131′ and 134, 134′ of the flowtubes 130 and 130′ arefixedly attached to flowtube mounting blocks 120 and 120′ and theseblocks, in turn, are fixedly attached to the manifolds 150 and 150′.This provides a continuous closed material path through the flowmeterassembly 10.

When the flanges 103 and 103′, having holes 102 and 102′ are connected,via the inlet end 104 and the outlet end 104′ into a process line (notshown) which carries the process material that is being measured,material enters the end 104 of the meter through an orifice 101 in theflange 103 and is conducted through the manifold 150 to the flowtubemounting block 120 having a surface 121. Within the manifold 150 thematerial is divided and routed through the flowtubes 130 and 130′. Uponexiting the flowtubes 130 and 130′, the process material is recombinedin a single stream within the manifold 150′ and is thereafter routed tothe exit end 104′ connected by the flange 103′ having bolt holes 102′ tothe process line (not shown).

The flowtubes 130 and 130′ are selected and appropriately mounted to theflowtube mounting blocks 120 and 120′ so as to have substantially thesame mass distribution, moments of inertia, and Young's modulus aboutthe bending axes W-W and W′-W′, respectively. These bending axes gothrough the brace bars 140 and 140′. Inasmuch as the Young's modulus ofthe flowtubes change with temperature, and this change affects thecalculation of flow and density, the resistive temperature detector(RTD) 190 is mounted to the flowtube 130′, to continuously measure thetemperature of the flowtube. The temperature dependent voltage appearingacross the RTD 190 may be used by the meter electronics 20 to compensatefor the change in the elastic modulus of the flowtubes 130 and 130′ dueto any changes in flowtube temperature. The RTD 190 is connected to themeter electronics 20 by the lead 195.

The first and second drivers 180L and 180R are spaced apart and arelocated at upstream and downstream portions of the flowtubes 130 and130′. A suitable drive signal is supplied to the first and seconddrivers 180L and 180R by the meter electronics 20 via the leads 185L and185R. The first and second drivers 180L and 180R may comprise any one ofmany well-known arrangements, such as a magnet mounted to the flowtube130′ and an opposing coil mounted to the flowtube 130 and through whichan alternating current is passed for vibrating both flowtubes 130, 130′.Depending on the polarity of the drive signal applied to the coilcomponent of the driver, a magnetic field can be generated which adds toor opposes the magnetic field of the magnet component of the driver. Asa result, the polarity of the drive signal can push the coil and magnetcomponents apart, causing the drive to expand, or can pull the coil andmagnet components together, causing the driver to contract. Theexpansion or contraction of the driver can move the flowtubes 130 and130′ apart or together.

The flowtubes 130 and 130′ may be driven by the first and second drivers180L and 180R in any desired vibration mode. In a bending mode (see FIG.4 and the accompanying discussion), the flowtubes 130 and 130′ may bedriven by a bending mode drive signal or signals in opposite directionsabout their respective bending axes W and W′ in what is termed the firstout-of-phase bending mode of the vibratory flowmeter 5. In a bendingmode vibration, the first and second drivers 180L and 180R are driven bythe drive signal or signals to operate synchronously and in phase, withthe first and second drivers 180L and 180R expanding simultaneously topush the flowtubes 130 and 130′ apart, and then will contractsimultaneously to pull the flowtubes 130 and 130′ together.

In a twist mode vibration (see FIG. 5 and the accompanying discussion),the first and second drivers 180L and 180R are driven by a twist modedrive signal to operate 180 degrees out of phase, with one driverexpanding and the other driver simultaneously contracting, wherein theupstream portion of the flowtubes 130 and 130′ will move apart while thedownstream portion will move together at one instance in time, and thenthe motion is reversed. As a result, the flowtubes 130 and 130′ includecentral nodes N and N′, wherein the flowtubes 130 and 130′ vibrate(i.e., twist) around the central nodes N and N′.

The meter electronics 20 receives the RTD temperature signal on the lead195, and the left and right velocity signals appearing on the leads 165Land 165R, respectively. The meter electronics 20 produces the drivesignal appearing on the leads 185L and 185R to the first and seconddrivers 180L and 180R and vibrates the flowtubes 130 and 130′. The meterelectronics 20 processes the left and right velocity signals and the RTDsignal to compute the mass flow rate and the density of the materialpassing through the flowmeter assembly 10. This information, along withother information, is applied by the meter electronics 20 over thecommunication link 26 to an external device or devices.

Flowmeters are inevitably affected by operation, by the operatingenvironment, and by the flow material flowing through the flowmeter. Asa result, the meter stiffness may change over time, such as due toerosion by the flow material, and corrosion, for example. Changes in themeter stiffness can result in erroneous flow rate measurements.Consequently, operating the vibratory flowmeter using a flow calibrationfactor value that was obtained at the time of manufacture may result inincreasingly inaccurate measurements by the vibratory flowmeter.

FIG. 2 shows meter electronics 20 for meter verification of thevibratory flowmeter 5 according to an embodiment of the invention. Themeter electronics 20 can include an interface 201 and a processingsystem 203. The meter electronics 20 receives and processes first andsecond sensor signals from the flowmeter assembly 10, such as pickoffsensor signals from the first and second pickoff sensors 170L, 170R.

The interface 201 transmits a drive signal or drive signals to thedrivers 180L and 180R via the leads 165L and 165R. The interface 201 cantransmit one drive signal to the two drivers 180L and 180R via the leads165L and 165R. Alternatively, the interface 201 can transmit twoseparate drive signals to the drivers 180L and 180R via the leads 165Land 165R. The two separate drive signals can be the same or can differfrom each other.

Alternatively, the interface 201 can transmit a drive signal or signalsand a meter verification excitation signal or signals to the drivers180L and 180R. As a result, the meter electronics 20 can injectadditional signals (i.e., meter verification excitation signals) intothe drivers 180L and 180R for the meter verification process. Primarymode currents and secondary mode currents can then be measured for thedrivers 180L and 180R due to the meter verification excitation signals.

The interface 201 receives the first and second sensor signals from thefirst and second pickoff sensors 170L and 170R via the leads 100 ofFIG. 1. The interface 201 can perform any necessary or desired signalconditioning, such as any manner of formatting, amplification,buffering, etc. Alternatively, some or all of the signal conditioningcan be performed in the processing system 203.

In addition, the interface 201 can enable communications between themeter electronics 20 and external devices, such as via the communicationlink 26, for example. The interface 201 can transfer measurement data toexternal devices via the communication link 26 and can receive commands,updates, data, and other information from external devices. Theinterface 201 can be capable of any manner of electronic, optical, orwireless communication.

The interface 201 in one embodiment includes a digitizer (not shown),wherein the sensor signal comprises an analog sensor signal. Thedigitizer samples and digitizes the analog sensor signal and produces adigital sensor signal. The interface/digitizer can also perform anyneeded decimation, wherein the digital sensor signal is decimated inorder to reduce the amount of signal processing needed and to reduce theprocessing time.

The processing system 203 conducts operations of the meter electronics20 and processes flow measurements from the flowmeter assembly 10. Theprocessing system 203 executes an operational routine 210 and therebyprocesses the flow measurements in order to produce one or more flowcharacteristics (or other flow measurements).

The processing system 203 can comprise a general purpose computer, amicroprocessing system, a logic circuit, or some other general purposeor customized processing device. The processing system 203 can bedistributed among multiple processing devices. The processing system 203can include any manner of integral or independent electronic storagemedium, such as the storage system 204. The storage system 204 may becoupled to the processing system 203 or may be integrated into theprocessing system 203.

The storage system 204 can store information used for operating thevibratory flowmeter 5, including information generated during theoperation of the vibratory flowmeter 5. The storage system 204 can storeone or more signals that are used for vibrating the flowtubes 130 and130′ and that are provided to the first and second drivers 180L and180R. In addition, the storage system 204 can store vibrational responsesignals generated by the first and second pickoff sensors 170L and 170Rwhen the flowtubes 130 and 130′ are vibrated.

The one or more drive signals may include drive signals for generating aprimary mode vibration and a secondary mode vibration, along with themeter verification excitation signals (tones), for example. The primarymode vibration in some embodiments may comprise a bending mode vibrationand the secondary mode vibration in some embodiments may comprise atwist mode vibration. However, other or additional vibration modes arecontemplated and are within the scope of the description and claims.

The meter electronics 20 can control the first and second drivers 180Land 180R to operate in a correlated manner, wherein the first and seconddrivers 180L and 180R receive drive signals that are substantiallyidentical in drive signal phase, drive signal frequency, and drivesignal amplitude. If the first and second drivers 180L and 180R areoperated in a correlated manner, then the stiffness and residualflexibility values comprise [2×1] vectors or matrices.

Alternatively, the meter electronics 20 can control the first and seconddrivers 180L and 180R to operate in an uncorrelated manner, wherein thefirst and second drivers 180L and 180R can differ during operation inone or more of drive signal phase, drive signal frequency, or drivesignal amplitude. If the first and second drivers 180L and 180R areoperated in an uncorrelated manner, then the stiffness and residualflexibility values comprise [2×2] vectors or matrices, generating twoadditional diagnostics for each of the stiffness and residualflexibility.

The storage system 204 can store a primary mode current 230. The primarymode drive current 230 may comprise a drive/excitation current orcurrents used to generate the primary vibration mode in the flowmeterassembly 10 as well as the meter verification signals. The primary modedrive current 230 may comprise currents from one or both of the firstand second drivers 180L and 180R. In some embodiments, the storagesystem 204 can store first and second primary mode currents 230corresponding to the first and second drivers 180L and 180R. The firstand second primary mode currents 230 can comprise commanded currents forthe primary vibration mode (i.e., the currents stipulated for the firstand second drivers 180L and 180R) or can comprise measured currents ofthe primary vibration mode (i.e., the currents measured as actuallyflowing through the first and second drivers 180L and 180R).

The storage system 204 can store a secondary mode current 236. Thesecondary mode current 236 may comprise a drive/excitation current orcurrents used to generate the secondary vibration mode in the flowmeterassembly 10 as well as the meter verification signals. The secondarymode current 236 may comprise currents from one or both of the first andsecond drivers 180L and 180R. In some embodiments, the storage system204 can store first and second secondary mode currents 236 correspondingto the first and second drivers 180L and 180R. The first and secondsecondary mode currents 236 can comprise commanded currents for thesecondary vibration mode or can comprise measured currents of thesecondary vibration mode.

The storage system 204 can store a primary mode response voltage 231.The primary mode response voltage 231 may comprise sinusoidal voltagesignals or voltage levels generated in response to the primary vibrationmode. The primary mode response voltage 231 may comprise voltage signalsor voltage levels (such as peak voltages) generated by one or both ofthe first and second pickoff sensors 170L and 170R. The responsevoltages will also include the responses at the meter verificationexcitation signal frequencies. In some embodiments, the storage system204 can store first and second primary mode response voltages 231corresponding to the first and second pickoff sensors 170L and 170R.

The storage system 204 can store secondary mode response voltage 237.The secondary mode response voltage 237 may comprise sinusoidal voltagesignals or voltage levels generated in response to the secondaryvibration mode. The secondary mode response voltage 237 may comprisevoltage signals or voltage levels (such as peak voltages) generated byone or both of the first and second pickoff sensors 170L and 170R. Theresponse voltages will also include the responses at the meterverification excitation signal frequencies. In some embodiments, thestorage system 204 can store first and second secondary mode responsevoltages 237 corresponding to the first and second pickoff sensors 170Land 170R.

The storage system 204 can store a meter stiffness value 216. The meterstiffness value 216 comprises a stiffness value that is determined fromvibrational responses generated during operation of the vibratoryflowmeter 5. The meter stiffness value 216 may be generated in order toverify proper operation of the vibratory flowmeter 5. The meterstiffness value 216 may be generated for a verification process, whereinthe meter stiffness value 216 serves the purpose of verifying proper andaccurate operation of the vibratory flowmeter 5.

The meter stiffness value 216 may be generated from the information ormeasurements generated during a primary vibration mode, during asecondary vibration mode, or both. Likewise, the residual flexibilityvalue may be generated from the information or measurements generatedduring a primary vibration mode, during a secondary vibration mode, orboth. If the meter stiffness value 216 is generated using informationfrom both the primary and secondary modes, then the meter stiffnessvalue 216 may be more accurate and reliable than if only one vibrationmode is used. When both the primary and secondary vibration modes areused, then a stiffness vector or matrix can be generated for each mode.Likewise, when both the primary and secondary vibration modes are used,then a residual flexibility vector or matrix can be generated for eachmode.

The vibrational response of a flowmeter can be represented by an openloop, second order drive model, comprising:M{umlaut over (x)}+C{dot over (x)}+Kx=f(t)  (3)where f is the force applied to the system, M is a mass parameter of thesystem, C is a damping parameter, and K is a stiffness parameter. Theterm x is the physical displacement distance of the vibration, the term{dot over (x)} is the velocity of the flowtube displacement, and theterm {umlaut over (x)} is the acceleration. This is commonly referred toas the MCK model. This formula can be rearranged into the form:(ms ² +cs+k)X(s)=F(s)+(ms+c)x(0)+m{dot over (x)}(0)  (4)

Equation (4) can be further manipulated into a transfer function form,while ignoring the initial conditions. The result is:

$\begin{matrix}{{H(s)} = {\frac{output}{input} = {\frac{X(s)}{F(s)} = \frac{\frac{1}{m}}{s^{2} + {\frac{c}{m}s} + \frac{k}{m}}}}} & (5)\end{matrix}$

Further manipulation can transform equation (5) into a first orderpole-residue frequency response function form, comprising:

$\begin{matrix}{{H(\omega)} = {\frac{R}{\left( {{j\;\omega} - \lambda} \right)} + \frac{\overset{\_}{R}}{\left( {{j\;\omega} - \overset{\_}{\left. \lambda \right)}} \right.}}} & (6)\end{matrix}$where λ is the pole, R is the residue, the term (j) comprises the squareroot of −1, and ω is the circular excitation frequency in radians persecond.

The system parameters comprising the natural/resonant frequency (ω_(n)),the damped natural frequency (ω_(d)), and the decay characteristic (ζ)are defined by the pole.

$\begin{matrix}{\omega_{n} = {\lambda }} & (7) \\{\omega_{d} = {{imag}(\lambda)}} & (8) \\{\zeta = \frac{{real}(\lambda)}{\omega_{n}}} & (9)\end{matrix}$

The stiffness parameter (K), the damping parameter (C), and the massparameter (M) of the system can be derived from the pole and residue.

$\begin{matrix}{M = \frac{1}{2j\; R\;\omega_{d}}} & (10) \\{K = {\omega_{n}^{2}M}} & (11) \\{C = {2{\zeta\omega}_{n}M}} & (12)\end{matrix}$

Consequently, the stiffness parameter (K), the mass parameter (M), andthe damping parameter (C) can be calculated based on a good estimate ofthe pole (λ) and the residue (R).

The pole and residue are estimated from the measured Frequency ResponseFunctions (FRFs). The pole (λ) and the residue (R) can be estimatedusing an iterative computational method, for example.

The response near the drive frequency is composed of primarily the firstterm of equation (6), with the complex conjugate term contributing onlya small, nearly constant “residual” part of the response. As a result,equation (6) can be simplified to:

$\begin{matrix}{{H(\omega)} = \frac{R}{\left( {{j\;\omega} - \lambda} \right)}} & (13)\end{matrix}$

In equation (13), the H(ω) term is the measured FRF. In this derivation,H is composed of a displacement output divided by a force input.However, with the voice coil pickoffs typical of a Coriolis flowmeter,the measured FRF (i.e., a {dot over (H)} term) is in terms of velocitydivided by force. Therefore, equation (13) can be transformed into theform:

$\begin{matrix}{{\overset{\Cup}{H}(\omega)} = {{{{H(\omega)} \cdot j}\;\omega} = \frac{j\;\omega\; R}{\left( {{j\;\omega} - \lambda} \right)}}} & (14)\end{matrix}$

Equation (14) can be further rearranged into a form that is easilysolvable for the pole (λ) and the residue (R).

$\begin{matrix}{{{\overset{\Cup}{H}\; j\;\omega} - {\overset{\Cup}{H}\;\lambda}} = {{j\;\omega\; R} =}} & (15) \\{\overset{\Cup}{H} = {R + {\frac{\overset{\Cup}{H}}{j\;\omega}\lambda}}} & (16) \\{{\begin{bmatrix}1 & \frac{\overset{\Cup}{H}}{j\;\omega}\end{bmatrix}\begin{Bmatrix}R \\\lambda\end{Bmatrix}} = \overset{\Cup}{H}} & (17)\end{matrix}$

Equations (15)-(17) form an over-determined system of equations.Equation (17) can be computationally solved in order to determine thepole (λ) and the residue (R) from the velocity/force FRF ({dot over(H)}). The terms H, R, and λ are complex.

Correlated drivers can be used in the primary mode, the secondary mode,or in multiple modes. In some embodiments, the drivers are correlatedand two FRFs may be measured in each of the primary and secondary modes.Consequently, four FRFs may be measured: 1) a FRF from the left driver180L to the left pickoff 170L, 2) a FRF from the left driver 180L to theright pickoff 170R), 3) a FRF from the right driver 180R to the leftpickoff 170L, and 4) a FRF from the right driver 180R to the rightpickoff 170R.

Recognizing that the FRFs share a common pole (λ) but separate residues(R_(L)) and (R_(R)), the two measurements can be combined advantageouslyto result in a more robust pole and residue determination.

$\begin{matrix}{{\begin{bmatrix}1 & 0 & \frac{{\overset{\Cup}{H}}_{LPO}}{j\;\omega} \\0 & 1 & \frac{{\overset{\Cup}{H}}_{RPO}}{j\;\omega}\end{bmatrix}\begin{Bmatrix}R_{L} \\R_{R} \\\lambda\end{Bmatrix}} = \overset{\Cup}{H}} & (18)\end{matrix}$

Equation (18) can be solved in any number of ways. In one embodiment,the equation is solved through a recursive least squares approach. Inanother embodiment, the equation is solved through a pseudo-inversetechnique. In yet another embodiment, because all of the measurementsare available simultaneously, a standard Q-R decomposition technique canbe used. The Q-R decomposition technique is discussed in Modern ControlTheory, William Brogan, copyright 1991, Prentice Hall, pp. 222-224,168-172.

After equation (18) is iteratively processed to a satisfactoryconvergence, then the pole and residue can be used for generatingstiffness values according to equations (10) and (11). With driverinputs that are correlated, Equations (10) and (11) can be used togenerate stiffness values between the drivers and the left pickoff andthe drivers and the right pickoff. In this case, the stiffness andresidual flexibility values for each mode are of the size [2×1].

Equations (10) and (11) can also be used to generate stiffness values Kbetween each pickoff sensor 170L and 170R and each driver 180L and 180R.Stiffness values generated can include a K_(LL) (auto) stiffness valuegenerated for the left pickoff sensor using the left driver, a K_(RL)(cross) stiffness value generated for the right pickoff sensor 170Rusing the left driver 180L, a K_(LR) (cross) stiffness value generatedfor the left pickoff sensor 170L using the right driver 180R, and aK_(RR) (auto) stiffness value generated for the right pickoff sensor170R using the right driver 180R. The two (auto) terms may be equal dueto the symmetry of the structure. The (cross) terms will always be equalto each other due to reciprocity, i.e., inputting a vibration at point Aand measuring the response at point B will generate the same vibrationalresponse result as inputting the vibration at point B and measuring theresponse at point A. The result is a stiffness matrix X:

$\begin{matrix}{X = \begin{bmatrix}K_{RR} & K_{LR} \\K_{RL} & K_{RR}\end{bmatrix}} & (19)\end{matrix}$

The stiffness matrix X can be stored as the meter stiffness value 216.

The storage system 204 can store a baseline meter stiffness 209 that isprogrammed into the meter electronics 20. In some embodiments, thebaseline meter stiffness 209 may be programmed into the meterelectronics 20 at the factory (or other manufacturer facility), such asupon construction or sale of the vibratory flowmeter 5. Alternatively,the baseline meter stiffness 209 may be programmed into the meterelectronics 20 during a field calibration operation or other calibrationor re-calibration operation. However, it should be understood that thebaseline meter stiffness 209 in most embodiments will not be changeableby a user or operator or during field operation of the vibratoryflowmeter 5.

If the meter stiffness value 216 is substantially the same as thebaseline meter stiffness 209, then it can be determined that thevibratory flowmeter 5 is relatively unchanged in condition from when itwas manufactured, calibrated, or when the vibratory flowmeter 5 was lastre-calibrated. Alternatively, where the meter stiffness value 216significantly differs from the baseline meter stiffness 209, then it canbe determined that the vibratory flowmeter 5 has been degraded and maynot be operating accurately and reliably, such as where the vibratoryflowmeter 5 has changed due to metal fatigue, corrosion, erosion due toflow, or other operating condition or effect.

The storage system 204 can store a predetermined stiffness range 219.The predetermined stiffness range 219 comprises a selected range ofacceptable stiffness values. The predetermined stiffness range 219 maybe chosen to account for normal wear on the vibratory flowmeter 5. Thepredetermined stiffness range 219 may be chosen to account for corrosionor erosion in the vibratory flowmeter 5.

In one embodiment, the storage system 204 stores a meter residualflexibility value 218. The meter residual flexibility value 218comprises a residual flexibility value that is determined fromvibrational responses generated during operation of the vibratoryflowmeter 5. Determining the residual flexibility only requiresadditional curve fitting during the stiffness calculation, requiringonly an additional iteration of the fitting algorithm or process forequation (18) in some embodiments. The residual flexibility has the sameform as the stiffness matrix (see equation (19) and the accompanyingdiscussion).

FIG. 3 is a graph of three FRFs showing the effect of residualflexibility, plotted as amplitude (A) versus frequency (f). Theamplitude peak of FRF₁ occurs at the first resonance frequency ω₁. Theamplitude peaks FRF₂ and FRF₃ occur at the resonance frequencies ω₂ andω₃. It can be seen from the graph that FRF₂ and FRF₃ have tails thataffect the amplitude values of FRF₁, including at the resonancefrequency ω₁. This effect of the tails of FRF₂ and FRF₃ on the vibrationat the resonance frequency ω₁ is called residual flexibility. Similarly,FRF₂ shows the residual flexibility effect of the tail of FRF₃.

Referring again to FIG. 2, the meter residual flexibility value 218 maybe generated in order to verify proper operation of the vibratoryflowmeter 5. The meter residual flexibility value 218 may be generatedfor a verification process, wherein the meter residual flexibility value218 serves the purpose of verifying proper and accurate operation of thevibratory flowmeter 5. When both the primary and secondary vibrationmodes are used, then a stiffness vector or matrix can be generated foreach mode. Likewise, when both the primary and secondary vibration modesare used, then a residual flexibility vector or matrix can be generatedfor each mode.

The above development assumed that the four FRFs are measuredsimultaneously, ignoring the need to sustain the meter at resonance, acondition of normal flow measurement operation. The need to sustainresonance complicates the issue in that four independent FRFs cannot besimultaneously measured in order to solve the problem. Rather, whencomputing FRFs, the aggregate effect of both drivers on the output canbe measured.

$\begin{matrix}{{\overset{\Cup}{H}}_{S_{-}} \equiv \frac{{\overset{.}{x}}_{L_{-}} + {\overset{.}{x}}_{R_{-}}}{f_{L} + f_{R}}} & (20)\end{matrix}$

In this equation, the {dot over (x)}_(L_) term refers to the velocity atthe selected pickoff due to the force at the left driver 180L and the{dot over (x)}_(R_) term refers to the velocity at the selected pickoffdue to the force at the right driver 180R. This quantity cannot bedirectly measured. Rather, only the sum of the two drivers' effects atthe pickoffs is measured. However, this quantity will be used in thetheoretical development that follows. The summed-effect FRF defined inequation (20) is insufficient to solve for the desired four residues.However, it can be solved with one more piece of information, the FRFbetween the driver forces.

$\begin{matrix}{H_{f} \equiv \frac{f_{L}}{f_{R}}} & (21)\end{matrix}$

To see how these two pieces of information are sufficient to solve thesystem model, the definition of the frequency response function for anarbitrary driver “D” is used to define:{dot over (x)} _(D_) ≡H̆ _(D_) f _(D)  (22)

Using linearity, the effects of equation (22) can be summed as appliedto the left and right drivers.{dot over (x)} _(L_) +{dot over (x)} _(R_) =H̆ _(L_) +f _(L) +H̆ _(R_) f_(R)  (23)

Both sides of equation (23) can be divided by any nonzero quantity. Forexample, equation (23) can be divided by the sum of the left and rightdriver forces, which are nonzero so long as the structure is beingexcited.

$\begin{matrix}{\frac{{\overset{.}{x}}_{L_{-}} + {\overset{.}{x}}_{R_{-}}}{f_{L} + f_{R}} = {{{\overset{\Cup}{H}}_{L_{-}}\left( \frac{f_{L}}{f_{L} + f_{R}} \right)} + {{\overset{\Cup}{H}}_{R_{-}}\left( \frac{f_{R}}{f_{L} + f_{R}} \right)}}} & (24)\end{matrix}$

The left-hand side of equation (24) can be directly measured. Theright-hand side features the individual FRFs that relate to the pole andresidues. The force ratios of equation (21) can be used to transformequation (24).

$\begin{matrix}{\frac{f_{L}}{f_{L} + f_{R}} = {\frac{f_{L}/f_{R}}{{f_{L}/f_{R}} + {f_{R}/f_{R}}} = {\frac{H_{f}}{H_{f} + 1} \equiv \gamma_{L}}}} & (25) \\{\frac{f_{R}}{f_{L} + f_{R}} = {\frac{f_{R}/f_{R}}{{f_{L}/f_{R}} + {f_{R}/f_{R}}} = {\frac{1}{H_{f} + 1} \equiv \gamma_{R}}}} & (26)\end{matrix}$

Note that the γ_(L) and γ_(R) terms are defined in the equations tofollow. Intuitively, though, they are the fraction of the total forceapplied at a particular driver. If the two drivers are driven equally,the γ_(L) and γ_(R) values are both 0.5. If one driver is driven fully,they are 0 and 1. In general, the γ_(L) and γ_(R) terms can be complexnumbers with a magnitude and phase relationship and are computed frommeasured force (or electrical driver current) FRFs.

Substituting equations (20), (25), and (26) into equation (24) yields:

$\begin{matrix}{{\overset{\Cup}{H}}_{S_{-}} = {\frac{{\overset{.}{x}}_{L_{-}} + {\overset{.}{x}}_{R_{-}}}{f_{L} + f_{R}} = {{\gamma_{L}{\overset{\Cup}{H}}_{L_{-}}} + {\gamma_{R}{\overset{\Cup}{H}}_{R_{-}}}}}} & (27)\end{matrix}$

The last step is to replace the system FRFs H̆_(L_) and H̆_(R_) with poleresidue models and rearrange the terms.

$\begin{matrix}{{{\gamma_{L}R_{L_{-}}} + {\gamma_{R}R_{R_{-}}} + {\frac{{\overset{\Cup}{H}}_{S_{-}}}{j\;\omega}\lambda}} = {\overset{\Cup}{H}}_{S_{-}}} & (28)\end{matrix}$

The gamma values and summed-FRFs in equation (28) are derived frommeasured data and are both functions of frequency. This basic equationcan be expanded over five tones that may be driven for meterverification and over the two pickoffs, giving a system with tenequations and five unknowns. For clarity this expansion is shown inequation (29). Once this system of equations has been used to solve forthe system parameters (R_(LL), R_(LR), R_(RL), R_(RR), λ), extractingthe stiffness vector or matrix is a trivial matter.

$\begin{matrix}{{\begin{bmatrix}{\gamma_{L}\left( \omega_{1} \right)} & {\gamma_{L}\left( \omega_{Dr} \right)} & 0 & 0 & \frac{{\overset{\Cup}{H}}_{SL}\left( \omega_{1} \right)}{j\;\omega_{1}} \\{\gamma_{L}\left( \omega_{2} \right)} & {\gamma_{R}\left( \omega_{2} \right)} & 0 & 0 & \frac{{\overset{\Cup}{H}}_{SL}\left( \omega_{2} \right)}{j\;\omega_{2}} \\{\gamma_{L}\left( \omega_{2} \right)} & {\gamma_{R}\left( \omega_{3} \right)} & 0 & 0 & \frac{{\overset{\Cup}{H}}_{SL}\left( \omega_{3} \right)}{j\;\omega_{3}} \\{\gamma_{L}\left( \omega_{4} \right)} & {\gamma_{R}\left( \omega_{4} \right)} & 0 & 0 & \frac{{\overset{\Cup}{H}}_{SL}\left( \omega_{4} \right)}{j\;\omega_{4}} \\{\gamma_{L}\left( \omega_{Dr} \right)} & {\gamma_{R}\left( \omega_{Dr} \right)} & 0 & 0 & \frac{{\overset{\Cup}{H}}_{SL}\left( \omega_{Dr} \right)}{j\;\omega_{Dr}} \\0 & 0 & {\gamma_{L}\left( \omega_{1} \right)} & {\gamma_{R}\left( \omega_{1} \right)} & \frac{{\overset{\Cup}{H}}_{SR}\left( \omega_{1} \right)}{j\;\omega_{1}} \\0 & 0 & {\gamma_{L}\left( \omega_{2} \right)} & {\gamma_{R}\left( \omega_{2} \right)} & \frac{{\overset{\Cup}{H}}_{SR}\left( \omega_{2} \right)}{j\;\omega_{2}} \\0 & 0 & {\gamma_{L}\left( \omega_{2} \right)} & {\gamma_{R}\left( \omega_{3} \right)} & \frac{{\overset{\Cup}{H}}_{SR}\left( \omega_{3} \right)}{j\;\omega_{3}} \\0 & 0 & {\gamma_{L}\left( \omega_{4} \right)} & {\gamma_{R}\left( \omega_{4} \right)} & \frac{{\overset{\Cup}{H}}_{SR}\left( \omega_{4} \right)}{j\;\omega_{4}} \\0 & 0 & {\gamma_{L}\left( \omega_{Dr} \right)} & {\gamma_{R}\left( \omega_{Dr} \right)} & \frac{{\overset{\Cup}{H}}_{SR}\left( \omega_{Dr} \right)}{j\;\omega_{Dr}}\end{bmatrix}\begin{Bmatrix}R_{LL} \\R_{RL} \\R_{LR} \\R_{RR} \\\lambda\end{Bmatrix}} = \left\lbrack \begin{matrix}{{\overset{\Cup}{H}}_{SL}*\left( \omega_{1} \right)} \\{{\overset{\Cup}{H}}_{SL}*\left( \omega_{2} \right)} \\{{\overset{\Cup}{H}}_{SL}*\left( \omega_{3} \right)} \\{{\overset{\Cup}{H}}_{SL}*\left( \omega_{4} \right)} \\{{\overset{\Cup}{H}}_{SL}*\left( \omega_{Dr} \right)} \\{{\overset{\Cup}{H}}_{SR}*\left( \omega_{1} \right)} \\{{\overset{\Cup}{H}}_{SR}*\left( \omega_{2} \right)} \\{{\overset{\Cup}{H}}_{SR}*\left( \omega_{3} \right)} \\{{\overset{\Cup}{H}}_{SR}*\left( \omega_{4} \right)} \\{{\overset{\Cup}{H}}_{SR}*\left( \omega_{Dr} \right)}\end{matrix} \right\rbrack} & (29)\end{matrix}$

The pole-residue model can be modified to include a single residualflexibility term to account for the aggregate effect of the other modes.This effect is assumed to be constant with frequency within the localmeasurements near the drive mode. This will be true if all other modesare higher-frequency than the drive mode and are sufficiently far awayto be treated as a pure stiffness. The modified pole-residue model is:

$\begin{matrix}{{H(\omega)} = {\frac{R}{{j\;\omega} - \lambda} + \Phi}} & (30)\end{matrix}$The model can be converted to a velocity FRF and the terms can berearranged to obtain the more readily solvable form:

$\begin{matrix}{{\overset{\Cup}{H}(\omega)} = {\frac{j\;\omega\; R}{{j\;\omega} - \lambda} + {j\;{\omega\Phi}}}} & (31)\end{matrix}$This model can be transformed into:

$\begin{matrix}{{\overset{\Cup}{H}}_{S_{-}} = {{\gamma_{L}R_{L_{-}}} + {{\gamma_{L}\left( {{j\;\omega} - \lambda} \right)}\Phi_{L_{-}}} + {\gamma_{R}R_{R_{-}}} + {{\gamma_{R}\left( {{j\;\omega} - \lambda} \right)}\Phi_{R_{-}}} + {\frac{{\overset{\Cup}{H}}_{S_{-}}}{j\;\omega}\lambda}}} & (32)\end{matrix}$

The equation is no longer strictly linear in terms of the unknowns, R,λ, and Φ. Rather, the Φ and λ terms are interdependent. This can behandled via simple iterative solution technique. The model is firstsolved without residual flexibility terms (using equation (28)), thenre-solved using the original estimate of the pole for the multipliers ofΦ. This approach works reasonably well because the pole estimate isfairly insensitive to the relatively small residual flexibility, muchmore so than the residues are. Since a new pole estimate is producedeach time equation (32) is evaluated, the iterative technique can berepeated until the pole stabilizes (although a single iteration may besufficient in practice). In an online implementation, where systemparameters are computed for a number of sequential measurements in time,it may be more useful or efficient to seed the estimate of the pole withthe value from the previous time window, rather than starting fromscratch with the model without residual flexibility each time.

For actual use, equation (32) can be expanded in the same way equation(28) was expanded into equation (29). With the addition of the residualflexibilities, which are also unique for each input/output pairing,there are now ten equations and nine unknowns. The system of equationsis not nearly as overdetermined as it was in the original meterverification, but experimental data has shown the results to still berelatively stable. These equations can be expanded by the addition of alow frequency term accounting for the coil resistance.

In the development thus far, the γ quantities (derived from theleft-right force FRFs and essentially the fraction of whole input forceapplied at a particular driver) have been treated as measuredquantities. However, the distribution of input forces between the leftand right drivers is a design parameter for the algorithm. The FRFs arestill measured to detect any variation from what was commanded (e.g.,due to back-EMF driving current back into the current amplifiers), butin an ideal world the γ quantities would be constants chosen for theprocedure. The individual γ values can be viewed as components of aspatial force matrix Γ:

$\begin{matrix}{\Gamma = \begin{bmatrix}{\gamma_{L}\left( \omega_{1} \right)} & {\gamma_{L}\left( \omega_{2} \right)} & {\gamma_{L}\left( \omega_{3} \right)} & {\gamma_{L}\left( \omega_{4} \right)} & {\gamma_{L}\left( \omega_{DR} \right)} \\{\gamma_{R}\left( \omega_{1} \right)} & {\gamma_{R}\left( \omega_{2} \right)} & {\gamma_{R}\left( \omega_{3} \right)} & {\gamma_{R}\left( \omega_{4} \right)} & {\gamma_{R}\left( \omega_{DR} \right)}\end{bmatrix}} & (33)\end{matrix}$

Here rows correspond to different input locations and columns todifferent frequencies. The matrix can be reshaped to fit however manyfrequencies (or drivers) are in use. The choice of Γ is not entirelyarbitrary. For instance, driving all tones equally on each driver willcause the matrix in equation (29) to be ill-conditioned for aleast-squares solution (since columns 1 and 2 and 3 and 4 would beidentical). Increasing the spatial separation of the tones results inbetter numerical behavior when solving, since columns of the matrix aremore differentiated. In an effort to maximize this separation, thedesign parameters can comprise:

$\begin{matrix}{\Gamma = \begin{bmatrix}1 & 0 & 1 & 0 & {.5} \\0 & 1 & 0 & 1 & {.5}\end{bmatrix}} & (34)\end{matrix}$

Of course, the actual measured values will not be identically equal tothe above values. The tones are each given entirely to a particulardriver. The drive tone is evenly split between the drivers to help matchthe symmetric drive-tone mode shape and minimize the excitation of theresidual flexibilities of other modes (twisting-type modes are notexcited very well, though higher-frequency symmetric modes may be).

In one embodiment, the storage system 204 stores a baseline meterresidual flexibility 220. In some embodiments, the baseline meterresidual flexibility 220 may be programmed into the meter electronics 20at the factory (or other manufacturer facility), such as uponconstruction or sale of the vibratory flowmeter 5. Alternatively, thebaseline meter residual flexibility 220 may be programmed into the meterelectronics 20 during a field calibration operation or other calibrationor re-calibration operation. However, it should be understood that thebaseline meter residual flexibility 220 in most embodiments will not bechangeable by a user or operator or during field operation of thevibratory flowmeter 5.

In one embodiment, the storage system 204 stores a predeterminedresidual flexibility range 221. The predetermined residual flexibilityrange 221 comprises a selected range of acceptable residual flexibilityvalues. The predetermined residual flexibility range 221 may be chosento account for normal wear on the vibratory flowmeter 5. Thepredetermined residual flexibility range 221 may be chosen to accountfor corrosion or erosion in the vibratory flowmeter 5.

In some embodiments, the storage system 204 stores a verificationroutine 213. The verification routine 213, when executed by theprocessing system 203, can perform a verification process for thevibratory flowmeter 5. In some embodiments, the processing system 203when executing the verification routine 213 is configured to generate ameter stiffness value. In some embodiments, the processing system 203when executing the verification routine 213 is configured to generate ameter stiffness value and verify the proper operation of the vibratoryflowmeter using the meter stiffness value. In some embodiments, theprocessing system 203 when executing the verification routine 213 isconfigured to generate a meter residual flexibility value. In someembodiments, the processing system 203 when executing the verificationroutine 213 is configured to generate a meter residual flexibility valueand verify the proper operation of the vibratory flowmeter using themeter residual flexibility value.

In some embodiments, the processing system 203 when executing theverification routine 213 is configured to vibrate the flowmeter assembly10 in a primary vibration mode using the first and second drivers 180Land 180R, determine first and second primary mode currents 230 of thefirst and second drivers 180L and 180R for the primary vibration modeand determining first and second primary mode response voltages 231generated by the first and second pickoff sensors 170L and 170R for theprimary vibration mode, generate a meter stiffness value 216 using thefirst and second primary mode currents 230 and the first and secondprimary mode response voltages 231, and verify proper operation of thevibratory flowmeter 5 using the meter stiffness value 216.

In some embodiments, the first and second primary mode currents 230comprise commanded current levels. Alternatively, in other embodimentsthe first and second primary mode currents 230 comprise measured currentlevels.

In some embodiments, the second driver 180R is uncorrelated with thefirst driver 180L. Alternatively, in other embodiments the first andsecond drivers 180L and 180R are operated in a correlated manner.

In some embodiments, verifying proper operation of the vibratoryflowmeter 5 comprises comparing the meter stiffness value 216 to apredetermined stiffness range 219, generating a verification indicationfor the vibratory flowmeter 5 if the meter stiffness value 216 fallswithin the predetermined stiffness range 219, and generating averification failure indication for the vibratory flowmeter 5 if themeter stiffness value 216 does not fall within the predeterminedstiffness range 219.

In some embodiments, the processing system 203 when executing theverification routine 213 is configured to vibrate the flowmeter assembly10 in a secondary vibration mode using the first and second drivers 180Land 180R, determine first and second secondary mode currents 236 of thefirst and second drivers 180L and 180R for the secondary vibration modeand determining first and second secondary mode response voltages 237 ofthe first and second pickoff sensors 170L and 170R for the secondaryvibration mode, and generate the meter stiffness value 216 using one orboth of the first and second primary mode currents 230 and the first andsecond primary mode response voltages 231 or the first and secondsecondary mode currents 236 and the first and second secondary moderesponse voltages 237.

In some embodiments, the processing system 203 when executing theverification routine 213 is configured to generate a meter residualflexibility value 218 using the first and second primary mode currents230 and the first and second primary mode response voltages 231.

In some embodiments, the processing system 203 when executing theverification routine 213 is configured to generate a meter residualflexibility value 218 using the first and second primary mode currents230 and the first and second primary mode response voltages 231, comparethe meter residual flexibility value 218 to a predetermined residualflexibility range 221, generate a verification indication for thevibratory flowmeter 5 if the meter residual flexibility value 218 fallswithin the predetermined residual flexibility range 221, and generate averification indication for the vibratory flowmeter 5 if the meterresidual flexibility value 218 does not fall within the predeterminedresidual flexibility range 221.

In some embodiments, the processing system 203 when executing theverification routine 213 is configured to vibrate the flowmeter assembly10 in a secondary vibration mode using the first and second drivers 180Land 180R, determine first and second secondary mode currents 236 of thefirst and second drivers 180L and 180R for the secondary vibration modeand determining first and second secondary mode response voltages 237 ofthe first and second pickoff sensors 170L and 170R for the secondaryvibration mode, and generate a meter residual flexibility value 218using one or both of the first and second primary mode currents 230 andthe first and second primary mode response voltages 231 or the first andsecond secondary mode currents 236 and the first and second secondarymode response voltages 237.

The verification operation is significant because it enables the meterelectronics 20 to make a stiffness determination in the field, withoutperforming an actual flow calibration test. It enables a stiffnessdetermination without a calibration test stand or other specialequipment or special fluids. This is desirable because performing a flowcalibration in the field is expensive, difficult, and time-consuming.

FIG. 4 represents a vibratory flowmeter 5 having curved flowtubes 130and 130′ wherein the two parallel curved flowtubes 130 and 130′ arevibrated in a bending mode. The dashed lines in the figure show the restpositions of the two flowtubes 130 and 130′. In the bending mode, thetubes are vibrated with respect to the bending axes W-W and W′-W′.Consequently, the flowtubes 130 and 130′ move periodically away fromeach other (as shown by the curved arrows), then toward each other. Itcan be seen that each flowtube 130 and 130′ moves as a whole withrespect to the bending axes W-W and W′-W′.

FIG. 5 represents the vibratory flowmeter 5 wherein the two parallelcurved flowtubes 130 and 130′ are vibrated in a twist (or Coriolis)mode. The dashed lines in the figure show the rest positions of the twoflowtubes 130 and 130′. In the twist mode, the flowtubes at the left endin the figure are being forced together, while at the right end in thefigure the flowtubes are being forced apart (in a Coriolis modevibration, the twist is induced by Coriolis forces in reaction to adriven vibration, but may be simulated or induced by using two or moredrivers to force the twist vibration). As a result, each flowtube isbeing twisted about a center point or node, such as the nodes N and N′.Consequently, the ends of the flowtubes 130 and 130′ (or upstream anddownstream portions) periodically move toward and away from each other(as shown by the curved arrows).

FIG. 6 is a flowchart 600 of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention. In step 601, theflowmeter assembly of the vibratory flowmeter is vibrated in a primaryvibration mode to generate a primary mode vibrational response. Theprimary mode vibrational response comprises electrical signals generatedby the first and second pickoff sensors 170L and 170R.

In some embodiments, the primary vibration mode may comprise a bendingmode. However, it should be understood that the vibration could compriseother vibration modes, including a secondary vibration mode (see FIG. 8and the accompanying text below). It should also be understood thatvibrating the flowmeter assembly at the primary vibration mode maycomprise vibrating in a predetermined vibration mode and substantiallyat a resonance frequency for the predetermined vibration mode.

In step 602, the first and second primary mode currents and the firstand second primary mode response voltages are determined. The first andsecond primary mode currents are the electrical currents flowing throughthe two drivers. The first and second primary mode currents can comprisecommanded values of the currents or can comprise measured current valuesfor the two drivers.

The first and second primary mode response voltages are the responsevoltages generated by the first and second pickoff sensors. The firstand second primary mode response voltages can comprise voltagesgenerated with operating at or near a resonant frequency of the primaryvibration mode.

In step 603, a meter stiffness value is generated. The meter stiffnessvalue may be generated using the first and second primary mode currentsand the first and second primary mode response voltages, as previouslydiscussed.

In step 604, the newly-generated meter stiffness value is compared tothe baseline meter stiffness. If the meter stiffness value is within thepredetermined stiffness range, then the method branches to step 605. Ifthe meter stiffness value is not within the predetermined stiffnessrange, then the method branches to step 606.

The comparison may comprise determining a difference between the meterstiffness value and the baseline meter stiffness, wherein the differenceis compared to a predetermined stiffness range. The predeterminedstiffness range may comprise a stiffness range that includes expectedvariations in measurement accuracy, for example. The predeterminedstiffness range may delineate an amount of change in the meter stiffnessthat is expected and is not significant enough to generate averification failure determination.

The predetermined stiffness range may be determined in any manner. Inone embodiment, the predetermined stiffness range may comprise apredetermined tolerance range above and below the baseline meterstiffness. Alternatively, the predetermined stiffness range may bederived from a standard deviation or confidence level determination thatgenerates upper and lower range boundaries from the baseline meterstiffness, or using other suitable processing techniques.

In step 605, a verification indication is generated since the differencebetween the meter stiffness value and the baseline meter stiffness fellwithin the predetermined stiffness range. The meter stiffness istherefore determined to not have changed significantly. No furtheraction may need to be taken, although the result may be logged,reported, et cetera. The indication may include an indication to theuser that the baseline meter stiffness is still valid. The successfulverification indication signifies that the baseline meter stiffness isstill accurate and useful and that the vibratory flowmeter is stilloperating accurately and reliably.

In step 606, a verification failure indication is generated since thedifference between the meter stiffness value and the baseline meterstiffness has exceeded the predetermined stiffness range. The stiffnessof the meter is therefore determined to have changed significantly. Aspart of the verification failure indication, a software flag, visualindicator, message, alarm, or other indication may be generated in orderto alert the user that the flowmeter may not be acceptably accurate andreliable. In addition, the result may be logged, reported, et cetera.

FIG. 7 is a flowchart 700 of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention. In step 701, theflowmeter assembly of the vibratory flowmeter is vibrated in a primaryvibration mode to generate a primary mode vibrational response, aspreviously discussed.

In step 702, the first and second primary mode currents and the firstand second primary mode response voltages are determined, as previouslydiscussed.

In step 703, a meter residual flexibility value is generated. The meterresidual flexibility value may be generated using the first and secondprimary mode currents and the first and second primary mode responsevoltages, as previously discussed.

In step 704, the newly-generated meter residual flexibility value iscompared to a baseline meter residual flexibility. If the meter residualflexibility value is within the predetermined residual flexibilityrange, then the method branches to step 705. If the meter residualflexibility value is not within the predetermined residual flexibilityrange, then the method branches to step 706.

The comparison may comprise determining a difference between the meterresidual flexibility value and the baseline meter residual flexibility,wherein the difference is compared to the predetermined residualflexibility range. The predetermined residual flexibility range maycomprise a residual flexibility range that includes expected variationsin measurement accuracy, for example. The predetermined residualflexibility range may delineate an amount of change in the meterresidual flexibility that is expected and is not significant enough togenerate a verification failure determination.

The predetermined residual flexibility range may be determined in anymanner. In one embodiment, the predetermined residual flexibility rangemay comprise a predetermined tolerance above and below the baselinemeter residual flexibility. Alternatively, the predetermined residualflexibility range may be derived from a standard deviation or confidencelevel determination that generates upper and lower range boundaries fromthe baseline meter residual flexibility, or using other suitableprocessing techniques.

In step 705, a verification indication is generated since the differencebetween the meter residual flexibility value and the baseline meterresidual flexibility fell within the predetermined residual flexibilityrange. The meter residual flexibility is therefore determined to nothave changed significantly. No further action may need to be taken,although the result may be logged, reported, et cetera. The indicationmay include an indication to the user that the baseline meter residualflexibility is still valid. The successful verification indicationsignifies that the baseline meter residual flexibility is still accurateand useful and that the vibratory flowmeter is still operatingaccurately and reliably.

In step 706, a verification failure indication is generated since thedifference between the meter residual flexibility value and the baselinemeter residual flexibility has exceeded the predetermined residualflexibility range. The residual flexibility of the meter is thereforedetermined to have changed significantly. As part of the verificationfailure indication, a software flag, visual indicator, message, alarm,or other indication may be generated in order to alert the user that theflowmeter may not be acceptably accurate and reliable. In addition, theresult may be logged, reported, et cetera.

FIG. 8 is a flowchart 800 of a meter verification method for a vibratoryflowmeter according to an embodiment of the invention. In step 801, theflowmeter assembly of the vibratory flowmeter is vibrated in a primaryvibration mode to generate a primary mode vibrational response, aspreviously discussed.

In step 802, the first and second primary mode currents and the firstand second primary mode response voltages are determined, as previouslydiscussed.

In step 803, the flowmeter assembly is vibrated in a secondary vibrationmode to generate a secondary mode vibrational response. In someembodiments, the secondary mode vibrational response is generatedsimultaneously with the primary mode vibrational response.Alternatively, the secondary vibration mode may be alternated with theprimary vibration mode.

In some embodiments, the primary vibration mode may comprise a bendingmode and the secondary vibration mode may comprise a twist mode.However, it should be understood that the vibration could comprise othervibration modes.

In step 804, first and second secondary mode drive currents and firstand second secondary mode response voltages are determined.

In step 805, a meter stiffness value is generated, as previouslydiscussed. The meter stiffness value may be generated using the firstand second primary mode currents and the first and second primary moderesponse voltages. The meter stiffness value may be generated using thefirst and second secondary mode currents and the first and secondsecondary mode response voltages. The meter stiffness value may begenerated using both the first and second primary mode currents and thefirst and second primary mode response voltages and the first and secondsecondary mode currents and the first and second secondary mode responsevoltages.

In step 806, the newly-generated meter stiffness value is compared tothe baseline meter stiffness. If the meter stiffness value is within thepredetermined stiffness range, then the method proceeds to step 808. Ifthe meter stiffness value is not within the predetermined stiffnessrange, then the method branches to step 811, wherein a verificationfailure indication is generated.

In step 808, a meter residual flexibility value is generated, aspreviously discussed. The meter residual flexibility value may begenerated using the first and second primary mode currents and the firstand second primary mode response voltages. The meter residualflexibility value may be generated using the first and second secondarymode currents and the first and second secondary mode response voltages.The meter residual flexibility value may be generated using both thefirst and second primary mode currents and the first and second primarymode response voltages and the first and second secondary mode currentsand the first and second secondary mode response voltages.

When both the primary and secondary vibration modes are used, then astiffness vector or matrix can be generated for each mode. Likewise,when both the primary and secondary vibration modes are used, then aresidual flexibility vector or matrix can be generated for each mode.

In step 809, the newly-generated meter residual flexibility value iscompared to a baseline meter residual flexibility. If the meter residualflexibility value is within the predetermined residual flexibilityrange, then the method branches to step 810. If the meter residualflexibility value is not within the predetermined residual flexibilityrange, then the method branches to step 811.

In step 810, a verification indication is generated since the differencebetween the meter stiffness value and the baseline meter stiffness fellwithin the predetermined stiffness range and the difference between themeter residual flexibility value and the baseline meter residualflexibility fell within the predetermined residual flexibility range.Therefore, it can be determined that both the baseline meter stiffnessand the baseline meter residual flexibility have not changedsignificantly. No further action may need to be taken, although theresult may be logged, reported, et cetera. The indication may include anindication to the user that the baseline meter stiffness and thebaseline meter residual flexibility are still valid. The successfulverification indication signifies that the baseline meter stiffness andthe baseline meter residual flexibility are still accurate and usefuland that the vibratory flowmeter is still operating accurately andreliably.

In step 811, a verification failure indication is generated since eitherthe difference between the meter stiffness value and the baseline meterstiffness has exceeded the predetermined stiffness range, the differencebetween the meter residual flexibility value and the baseline meterresidual flexibility has exceeded the predetermined residual flexibilityrange, or both. One or both of the meter stiffness or the meter residualflexibility have changed significantly. As part of the verificationfailure indication, a software flag, visual indicator, message, alarm,or other indication may be generated in order to alert the user that theflowmeter may not be acceptably accurate and reliable. In addition, theresult may be logged, reported, et cetera.

The vibratory flowmeter and method according to any of the embodimentscan be employed to provide several advantages, if desired. The vibratoryflowmeter and method according to any of the embodiments quantifies theflowmeter stiffness using one or more vibration modes to generate animproved and more reliable meter stiffness value. The vibratoryflowmeter and method according to any of the embodiments quantifies theflowmeter residual flexibility using one or more vibration modes togenerate an improved and more reliable meter stiffness value. The meterstiffness analysis method may determine if the vibratory flowmeter isstill accurate and reliable.

The detailed descriptions of the above embodiments are not exhaustivedescriptions of all embodiments contemplated by the inventors to bewithin the scope of the invention. Indeed, persons skilled in the artwill recognize that certain elements of the above-described embodimentsmay variously be combined or eliminated to create further embodiments,and such further embodiments fall within the scope and teachings of theinvention. It will also be apparent to those of ordinary skill in theart that the above-described embodiments may be combined in whole or inpart to create additional embodiments within the scope and teachings ofthe invention. Accordingly, the scope of the invention should bedetermined from the following claims.

What is claimed is:
 1. A vibratory flowmeter (5) for meter verification,the vibratory flowmeter (5) comprising: a flowmeter assembly (10)including one or more flowtubes (130, 130′) and first and second pickoffsensors (170L, 170R); first and second drivers (180L, 180R) configuredto vibrate the one or more flowtubes (130, 130′); and meter electronics(20) coupled to the first and second pickoff sensors (170L, 170R) andcoupled to the first and second drivers (180L, 180R), with the meterelectronics (20) being configured to vibrate the flowmeter assembly (10)in a primary vibration mode using the first and second drivers (180L,180R), determine first and second primary mode currents (230) of thefirst and second drivers (180L, 180R) for the primary vibration mode,determine first and second primary mode response voltages (231)generated by the first and second pickoff sensors (170L, 170R) for theprimary vibration mode, and generate a meter stiffness value (216) usingthe first and second primary mode currents (230) and the first andsecond primary mode response voltages (231), the meter stiffness valuebeing a 2×2 matrix including a cross stiffness term K_(LR) and a crossstiffness term K_(RL), wherein the primary vibration mode is a naturalvibration mode of the one or more flow tubes.
 2. The vibratory flowmeter(5) of claim 1, with the first and second primary mode currents (230)comprising commanded current levels.
 3. The vibratory flowmeter (5) ofclaim 1, with the first and second primary mode currents (230)comprising measured current levels.
 4. The vibratory flowmeter (5) ofclaim 1, wherein the second driver (180R) is uncorrelated with the firstdriver (180L).
 5. The vibratory flowmeter (5) of claim 1, with the meterelectronics (20) being further configured to compare the meter stiffnessvalue (216) to a predetermined stiffness range (219), generate averification indication for the vibratory flowmeter (5) if the meterstiffness value (216) falls within the predetermined stiffness range(219), and generate a verification failure indication for the vibratoryflowmeter (5) if the meter stiffness value (216) does not fall withinthe predetermined stiffness range (219).
 6. The vibratory flowmeter (5)of claim 1, with the meter electronics (20) being further configured tovibrate the flowmeter assembly (10) in a secondary vibration mode usingthe first and second drivers (180L, 180R), determine first and secondsecondary mode currents (236) of the first and second drivers (180L,180R) for the secondary vibration mode and determine first and secondsecondary mode response voltages (237) of the first and second pickoffsensors (170L, 170R) for the secondary vibration mode, and generate themeter stiffness value (216) using one or both of the first and secondprimary mode currents (230) and the first and second primary moderesponse voltages (231) or the first and second secondary mode currents(236) and the first and second secondary mode response voltages (237).7. The vibratory flowmeter (5) of claim 1, with the meter electronics(20) being further configured to generate a meter residual flexibilityvalue (218) using the first and second primary mode currents (230) andthe first and second primary mode response voltages (231).
 8. Thevibratory flowmeter (5) of claim 1, with the meter electronics (20)being further configured to generate a meter residual flexibility value(218) using the first and second primary mode currents (230) and thefirst and second primary mode response voltages (231), compare the meterresidual flexibility value (218) to a predetermined residual flexibilityrange (221), generate a verification indication for the vibratoryflowmeter (5) if the meter residual flexibility value (218) falls withinthe predetermined residual flexibility range (221), and generate averification failure indication for the vibratory flowmeter (5) if themeter residual flexibility value (218) does not fall within thepredetermined residual flexibility range (221).
 9. The vibratoryflowmeter (5) of claim 1, with the meter electronics (20) being furtherconfigured to vibrate the flowmeter assembly (10) in a secondaryvibration mode using the first and second drivers (180L, 180R),determine first and second secondary mode currents (236) of the firstand second drivers (180L, 180R) for the secondary vibration mode anddetermining first and second secondary mode response voltages (237) ofthe first and second pickoff sensors (170L, 170R) for the secondaryvibration mode, and generate a meter residual flexibility value (218)using one or both of the first and second primary mode currents (230)and the first and second primary mode response voltages (231) or thefirst and second secondary mode currents (236) and the first and secondsecondary mode response voltages (237).